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Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifolds

Schulze, F; (2020) Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifolds. Geometric and Functional Analysis , 30 pp. 255-288. 10.1007/s00039-020-00522-8. Green open access

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Abstract

Let (Mn,g) be simply connected, complete, with non-positive sectional curvatures, and Σ a 2-dimensional closed integral current (or flat chain mod 2) with compact support in M. Let S be an area minimising integral 3-current (resp. flat chain mod 2) such that ∂S=Σ. We use a weak mean curvature flow, obtained via elliptic regularisation, starting from Σ, to show that S satisfies the optimal Euclidean isoperimetric inequality: 6π−−√M[S]≤(M[Σ])3/2. We also obtain an optimal estimate in case the sectional curvatures of M are bounded from above by −κ<0 and characterise the case of equality. The proof follows from an almost monotonicity of a suitable isoperimetric difference along the approximating flows in one dimension higher and an optimal estimate for the Willmore energy of a 2-dimensional integral varifold with first variation summable in L2.

Type: Article
Title: Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifolds
Open access status: An open access version is available from UCL Discovery
DOI: 10.1007/s00039-020-00522-8
Publisher version: https://doi.org/10.1007/s00039-020-00522-8
Language: English
Additional information: This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10089257
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